Exponents
Introduction Just as multiplication is shorthand for addition, there is a shorthand for multiplication as well: exponentiation. Suppose one wanted to multiply a certain number a by itself some number of times b . For example, if a = 2 and b = 3 then we have 2 \cdot 2 \cdot 2 = 8 . This is more concisely expressed as a^b . Here is what it may look like visually: a^b = \underbrace{a \cdot a \cdot ... \cdot a}_{b \; times} Here, the thing we wish to multiply, a , is called the base and the number of times, b , is called the exponent. Exercises Evaluate the following: 2^4 {10}^5 1^{6789} 4^2 (\frac{1}{3})^3 Addition of exponents One advantage you get is the ability to multiply certain quantities more easily. For example, what is 2^3 \cdot 2^4 ? If we write it out in full it may look like this: 2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) The parentheses are there for clarity. But really, we can combine it into one expression: (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) = 2^7 We have a pattern emerging. We can now say: \begin{array}{lcl} a^b \cdot a^c & = & \underbrace{a \cdot a \cdot ... \cdot a}_{b \; times} \cdot \underbrace{a \cdot a \cdot ... \cdot a}_{c \; times} \\ & = & \underbrace{a \cdot a \cdot ... \cdot a}_{b+c \; times} \\ & = & a^{b+c} \end{array} Or more simply: a^b \cdot a^c = a^{b+c} In other words, exponents add when we multiply two expressions with the same base. Exercises Simplify into one expression of the form a^b : 2^3 \cdot 2^2 5^{123} \cdot 5^{456} 3^4 \cdot 3^5 \cdot 3^6 It is important to note that sometimes, the results of an expression involving exponentiation becomes far too expensive to calculate or to write down. For example, {17}^{23} = 19967568900859523802559065713 . Even though one can slowly crunch out this figure, leaving the expression as {17}^{23} would suffice because the meaning is clear. (In fact, the calculation itself is often unimportant or unnecessary!) Multiplication of exponents If it is possible to add exponents as in the above, then it is possible to multiply them: \begin{array}{lcl} (a^b)^c & = & \underbrace{a^b \cdot a^b \cdot ... \cdot a^b}_{c \; times} \\ & = & a^{b \cdot c} \end{array} Exercises Simplify into one expression of the form a^b : (3^{100})^{100} 5^2 \cdot (5^4)^6 \cdot (5^8)^{10} ((2^3)^4)^5 Multiplication of bases We can not always combine exponents together so easily. For example, how do we multiply 2^5 and 3^4 ? There is no simple expression of the form a^b which this can become. We have no choice but to multiply them out separately. However, what if two exponents were equal? That is, what if we wanted to multiply a^c and b^c ? Another pattern emerges: \begin{array}{lcl} a^c \cdot b^c & = & \underbrace{a \cdot a \cdot ... \cdot a}_{c \; times} \cdot \underbrace{b \cdot b \cdot ... \cdot b}_{c \; times} \\ & = & \underbrace{(a \cdot b) \cdot (a \cdot b) ... \cdot (a \cdot b)}_{c \; times} \\ & = & (a \cdot b)^c \end{array} Or more simply: a^c \cdot b^c = (a \cdot b)^c Exercises Simplify into one expression of the form a^b : 2^3 \cdot 3^3 1^{50} \cdot 5^{50} 3^7 \cdot 4^7 \cdot 5^7 6^{10} \cdot (\frac{1}{2})^{10} Factoring of bases Bases multiply when exponents are equal. We can reverse the process and decompose a base into smaller factors, all with the same exponent. For example, since 6 = 2 \cdot 3 we can say that 6^a = 2^a \cdot 3^a . More can be done when the base has repeated prime factors. For example, since 24 = 2^3 \cdot 3 we can say that {24}^a = (2^3)^a \cdot 3^a = 2^{3a} \cdot 3^a by multiplying the exponent. Exercises Completely factor: 10^4 {30}^3 8^{50} {21}^5 \cdot {35}^7